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Stochastic Calculus,Spring 2013, Lecture 3
Reading for this lecture:(1) [1] pp
10!"11#
(2) [2] pp 2#"31(3) [3] pp $#"$%
&anilla 'arrier ptions n finance, a *arrier option is a t+pe
of contract here option toe-ercise at .aturit+ depends on the
underl+ing crossing or reaching a gi/en *arrier le/elhere are
se/eral t+pes of /anilla *arrier options (/anillastands for si.ple
and liuid on the.ar4et instru.ent) So.e 4noc4"out henthe underl+ing
asset price crosses a *arrier (ie,the+ *eco.e orthless) f
theunderl+ing asset price *egins *elo the *arrier and .ust crossthe
*arrier a*o/e itto cause the 4noc4"out, the option is said to *e
up"and"out 5 don"and"out optionhas the *arrier *elo the initial
asset price and 4noc4s out if the asset price falls*elo the *arrier
ther options 4noc4"in at a *arrier (ie, pa+off 6ero unless the+
cross a*arrier) 7noc4"in options also fall in to categories:
up"and"in anddon"and"in he pa+off ate-piration is usuall+ a fi-ed
a.ount, a call or a puthere are also .ore co.plicated
*arrieroptions, for instance, R' (range in *ound) options hich for
a specified financial inde-, forinstance 3".onth Li*or rate or 2"
+ear C8S rate, pa+s a fi-ed a.ount .ultiplied *+ thefraction of
o*ser/ations henthe inde- is inside a specified range
Later in this lecture e treat the si.plest case 9e price an
up"and"out optionhose pa+off ate-piration is a call and e assu.e
that the stoc4 price is .od" elled *+ 'ronian .otion
5ssu.ption that the stoc4 price can *e .odelled *+ 'ronian
.otion is rather unrealistic as'ronian .otion (e/en ith large
drift)can *e negati/e ith positi/e pro*a*ilit+ Later in the coursee
ill *e a*le toappl+ si.ilar calculations for the .ore realistic
case hen the asset price follosageo.etric 'ronian .otion
Running .a-i.u. and first passage ti.e or a stochastic process
;t, t 0e define
the running .a-i.u. as
8t< .a- ;
s (1)
0st
or this process to *e ell defined e assu.e that the process ;tis
has continuous
tra=ectories(in fact, e need onl+ ith pro*a*ilit+ one) Closel+
related torunning.a-i.u. is the first passage ti.e hich is defined
as
a< inf{t > 0 : ;
t< a}, (2)that is for
a fi-ed a ti.e a
is the first ti.e hen ;treaches le/el a ro. the
definitions it is clear that e/ents {a? t} and {8
t a} coincide, ie,
{a? t} {8
t> a} (3)
ro. the e-a.ples in the *eginning of the lecture one can
conclude that the /alueof .an+*arrier options depends on the
*eha/ior of the .a-i.u. asset price prior tooption e-piration,or
eui/alentl+, on the distri*ution of the running .a-i.u. orinstance,
the 4noledge of thedistri*ution of the running .a-i.u. (or
eui/alentl+ of the first passage ti.e) is enough to
co.pute the /alue of the si.ple 4noc4"in@outoptions1
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'ut it turns out that for a general stochastic process ;tthe
distri*ution of
the running .a-i.u. is rather co.plicated and often cannot
*eritten out in a closedfor.
'ronian .otion n the case hen stochastic process ;t
is a 'ronian
.otion e can e-plicitl+ calculate the distri*ution of the
running
.a-i.u. and hittingti.e
heore. 1 (Reflection Arinciple) Let a > 0 hen A(a? t) <
2A('
t
> a)
Re.ar4 2 'efore the start of the proof let us rerite the a*o/e
euation as
A(a? t) < 2
2r
e- @
(2t)
2Bta
d- ()
o find the pro*a*ilit+ densit+ of
ae change/aria*les - a} is a su*set of the e/ent {
a?
t},thus
A(a? t) < 2A(
a? t, '
t> a) < 2A('
t> a) (%)
E
Re.ar4 3 rul+ spea4ing, the fact that 't '
ais independent of F
a
(infor".ation a/aila*le at ti.e a) needs to *e pro/ed using
the
strong 8ar4o/ propert+for 'ronian .otion 9e refer to [2] for
the
proofne can go further and generali6e the result of 0heore. 1
Let u ? / a, thenusingthe reflection principle e can easil+ sho
that
A(a? t, u ? '
t? /) < A(2a / ? '
t? 2a u) (!)
Since e/ents {a? t} and {8
t> a} are eui/alent
e o*tain
A(8t> a, u ? '
t? /) < A(2a / ? '
t? 2a u) ($)
Fo let the inter/al (u, /) shrin4 to -, so
that1 2
A(8t> a, '
t< -) < A('
t< 2a -)
